Setwise Hierarchical Variable Selection and the Generalized Linear Step-Up Procedure for False Discovery Rate Control

Abstract

Controlling the false discovery rate (FDR) in variable selection becomes challenging when predictors are correlated, as existing methods often exclude all members of correlated groups and consequently perform poorly for prediction. We introduce a new setwise variable-selection framework that identifies clusters of potential predictors rather than forcing selection of a single variable. By allowing any member of a selected set to serve as a surrogate predictor, our approach supports strong predictive performance while maintaining rigorous FDR control. We construct sets via hierarchical clustering of predictors based on correlation, then test whether each set contains any non-null effects. Similar clustering and setwise selection have been applied in the familywise error rate (FWER) control regime, but previous research has been unable to overcome the inherent challenges of extending this to the FDR control framework. To control the FDR, we develop substantial generalizations of linear step-up procedures, extending the Benjamini-Hochberg and Benjamini-Yekutieli methods to accommodate the logical dependencies among these composite hypotheses. We prove that these procedures control the FDR at the nominal level and highlight their broader applicability. Simulation studies and real-data analyses show that our methods achieve higher power than existing approaches while preserving FDR control, yielding more informative variable selections and improved predictive models.